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A386831 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

%I A386831 #7 Aug 05 2025 10:01:38
%S A386831 1,23,814,32102,1330436,56734023,2464566064,108464237352,
%T A386831 4819668737436,215760575713148,9716002818365314,439628651114930102,
%U A386831 19971546503835844436,910318041046245082898,41611957337801849102064,1906855257451887625497852,87569968895543824193201436
%N A386831 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
%F A386831 a(n) = [x^n] (1+3*x)^(4*n+1)/(1-2*x)^(3*n+1).
%F A386831 a(n) = [x^n] 1/((1-3*x) * (1-5*x)^(3*n+1)).
%F A386831 a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(4*n+1,k).
%F A386831 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n+k,k).
%o A386831 (PARI) a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));
%Y A386831 Cf. A386829, A386830.
%Y A386831 Cf. A386765.
%K A386831 nonn
%O A386831 0,2
%A A386831 _Seiichi Manyama_, Aug 05 2025